Computing the minimum number of hybridization events for a consistent evolutionary history

Abstract

It is now well-documented that the structure of evolutionary relationships between a set of present-day species is not necessarily tree-like. The reason for this is that reticulation events such as hybridizations mean that species are a mixture of genes from different ancestors. Since such events are relatively rare, a fundamental problem for biologists is to determine the smallest number of hybridization events required to explain a given (input) set of data in a single (hybrid) phylogeny. The main results of this paper show that computing this smallest number is APX-hard, and thus NP-hard, in the case the input is a collection of phylogenetic trees on sets of present-day species. This answers a problem which was raised at a recent conference (Phylogenetic Combinatorics and Applications, Uppsala University, 2004). As a consequence of these results, we also correct a previously published NP-hardness proof in the case the input is a collection of binary sequences, where each sequence represents the attributes of a particular present-day species. The APX-hardness of these problems means that it is unlikely that there is an efficient algorithm for either computing the result exactly or approximating it to any arbitrary degree of accuracy.

Reconstruction of certain phylogenetic networks from the genomes at their leaves

“…Wang et al (2001) considered the problem of finding a perfect phylogenetic network with recombination that has the smallest number of recombination events. They suggested that the problem is NP-hard, and a full proof was given by Bordewich and Semple (2007). Wang et al then considered a restricted problem in which all recombination events are associated with nodedisjoint recombination cycles.…”

Efficiently Calculating Evolutionary Tree Measures Using SAT

“…That is, a forest that is maximal with respect to all agreement forests that have acyclic associated graphs has size equivalent to the hybridization number of the two trees [6]. See Figure 3.…”

“…Hardness Results: Both of these measures, hybridization number and rSPR distance have been shown to be NP-hard and fixed parameter tractable [5,6].…”

“…Bordewich and Semple [6] showed that the hybridization number is NPhard and fixed parameter tractable, by relating it with an appropriately defined agreement forest. Agreement forests were developed for evolutionary tree metrics in the pioneering work of Hein et al [14] and Allen and Steel [1] that linked the tree distance to the size of the maximum agreement forest (MAF).…”

A review of genetic analyses of hybridisation in New Zealand

“…The arrival of exotic species has been well documented and geological studies give us some ability to date the fragmentation, expansion and hybridisation of our native species. New Zealand mathematicians who are developing novel methods to study hybridisation will continue to give us impact in the international scientific community (for example Huson 2005;Winkworth et al 2005;Barom et al 2006;McBreen & Lockhart 2006;Bordewich & Semple 2007;Joly et al 2007, in press a; Holland et al 2008).…”